Math 2510 Spring 2010 -- Homework Problem Set #4

Due: Wednesday, March 17th
  1. Prove that \( \displaystyle{ \left\langle \frac{1}{n^3+12345} \right\rangle }\) converges to 0.

  2. Show that \(\displaystyle{\left\langle \frac{n^2+5}{4n^2} \right\rangle}\) where \( n > 0 \) converges to \(\displaystyle{\frac{1}{4} }\).

  3. Consider the sequence \(\displaystyle{ \left\langle (-1)^n n^2 \right\rangle }\). This sequence obviously diverges.

    1. Prove that this sequence diverges by showing it is unbounded.

    2. Prove that this sequence diverges by showing it does not converge to any real number \( L \).

  4. Suppose that \( \displaystyle{ \left\langle a_n \right\rangle } \) converges to \( L \). Show that \( \displaystyle{ \left\langle |a_n| \right\rangle } \) converges to \( |L| \).

  5. Suppose that \( \displaystyle{ \left\langle a_n \right\rangle } \) and \( \displaystyle{ \left\langle b_n \right\rangle } \) converge to 0. Show that \( \displaystyle{ \left\langle a_nb_n \right\rangle } \) converges to 0 as well.